Exchange Rate Economics: Theories and Evidence

The new open economy macroeconomics part 1 255
The model may be solved by using the demand curve (10.9) to substitute for p t (j)
in the budget constraint (10.4) then using the resulting expression to substitute for
C j
t in (10.1). This gives the unconstrained maximisation problem (the individual
takes C w as given):
max U j
y(j)M j B j t =
[
∞∑
β
{log
s−t (1 + r s )Fs
j
s=t
− τ s − F j
s+1 − M s
j ]
+ η log
P s
+ M j
s−1
P s
+ y s (j) (θ−1)/θ (C w s )1/θ
(
Ms
j ) }
− κ P s 2 y s(j) 2 . (10.10)
The first-order conditions derived from this maximising problem are:
C t+1 = β(1 + r t+1 )C t ,(10.11)
M t
P t
y (θ+1)/θ
= ηC t
( 1 + it+1
i t+1
) 1/ɛ
,(10.12)
t = θ − 1
θ κ
(C w
t ) 1/θ 1 C t
,(10.13)
where the j index has been suppressed and i t+1 is the nominal interest rate for
home country loans between t and t + 1. 3 Equation (10.11) is the familiar firstorder
consumption Euler equation for the composite consumption index (note the
inter-temporal elasticity of substitution is assumed to be unity). The first-order
condition for real money balances simply indicates that in equilibrium agents
should be indifferent between consuming a unit of consumption in period t or
using the same funds to raise cash balances,enjoying the derived transactions
utility in period t and converting this into consumption in period t + 1. The firstorder
condition for the labour–leisure trade-off in equation (10.13) ensures that the
marginal utility cost of producing an extra unit of output (in terms of the foregone
leisure) equals the marginal utility from consuming the added revenue that an extra
unit of output brings. 4
The technique adopted by Obstfeld and Rogoff (1995,1996) to solve the model
involves solving for the steady state and then examining the dynamic effects of
a monetary shock by taking a log-linear approximation around this steady state.
Before proceeding,it is worth noting that in the steady state,where consumption
and output are constant,the Euler equation (10.11) ties down the real interest
rate as: 5
¯r = δ ≡ 1 − β
β ,
where the overbar indicates a steady-state value.